Finite Group Databasebeta

← #27286806

Subgroups — #27286806

Order 13568.

588 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285868C211yesmaximal, characteristic
2#20285665C211yesmaximal, characteristic
3#20283113C211yesmaximal, characteristic
4#20275692C211yesmaximal, characteristic
5#20273399C211yesmaximal, characteristic
6#20273395C211yesmaximal, characteristic
7C53×#1276C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9C2×#13133217—22no—
10C2×#13133217C2211yescharacteristic
11C2×#13133217C2211yescharacteristic
12#16811028—21no—
13#16811003—21no—
14#16810974—21no—
15#16810974—21no—
16#16810959—22no—
17#16810919—22no—
18#16810834—22no—
19#16810781—22no—
20#16810647C2211yescharacteristic
21#16810493—22no—
22#16810493—22no—
23#16810383—22no—
24#16810371—22no—
25#16810370—22no—
26C53×#431C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C43×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×Dic106D412yes—
31C2×Dic212—24no—
32C2×#11719917—21no—
33#13133217—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217—21no—
41#13133217—48no—
42#13133217D411yescharacteristic
43#13133217D411yescharacteristic
44#13133209—21no—
45#13133196—21no—
46#13133196—21no—
47C4×Dic106—21no—
48C4×Dic106—21no—
49#13133176—21no—
50C2×C4⋊Dic53—28no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53D412yes—
55#13133151—24no—
56C4⋊#88322—21no—
57C4⋊#88322—21no—
58#13133142—24no—
59#13133142—24no—
60C2×C42×C53—21no—
61C2×C42×C53—21no—
62C2×C42×C53D412yes—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53D412yes—
65C4⋊C8×C53D412yes—
66C22⋊C8×C53—21no—
67C22⋊C8×C53—21no—
68C4×C8×C53—21no—
69C4×C8×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—22no—
75C22×Dic53—22no—
76C2×Dic106—21no—
77C2×Dic106—22no—
78C2×Dic106—21no—
79C2×Dic106—22no—
80C2×Dic106—44no—
81C2×Dic106—44no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C2×C8×C53—22no—
85C2×C8×C53—22no—
86C42×C53—24no—
87C42×C53—24no—
88C42×C53—22no—
89C42×C53Q1614yescomplemented
90C42×C53C2×D411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53C2×D411yescharacteristic
93C42×C53#4111yescharacteristic
94C22⋊Dic53—41no—
95C4⋊Dic53—21no—
96C4⋊Dic53—22no—
97C4⋊Dic53—21no—
98C4⋊Dic53—22no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—416no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107C4⋊Dic53—41no—
108#11719919—41no—
109#11719919—41no—
110C4×Dic53—41no—
111#11719917—42no—
112C2×#88322—41no—
113Dic212—84no—
114C23×C53#12011yescharacteristic
115C2×C4×C53—24no—
116C2×C4×C53—24no—
117C2×C4×C53—24no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53#12011yescharacteristic
121C2×C4×C53#12011yescharacteristic
122C2×C4×C53#12111yescharacteristic
123C2×C4×C53#12111yescharacteristic
124C2×C4×C53#12111yescharacteristic
125C2×C4×C53#12711yescharacteristic
126C2×C4×C53C2×Q1612yes—
127C2×C4×C53#13712yes—
128C2×Dic53—44no—
129C2×Dic53—44no—
130C2×Dic53—42no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—42no—
134Dic106—44no—
135Dic106—42no—
136Dic106—42no—
137Dic106—82no—
138Dic106—82no—
139Dic106—82no—
140C8×C53—42no—
141C8×C53—42no—
142#88322—81no—
143#10293—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53#39211yescharacteristic
147C22×C53#45112yes—
148C4×C53—28no—
149C4×C53—21no—
150C4×C53—21no—
151C4×C53#45312yes—
152C4×C53#45612yes—
153C4×C53#46212yes—
154C4×C53#49411yescharacteristic
155C4×C53#49711yescharacteristic
156Dic53—82no—
157Dic53—81no—
158Dic53—82no—
159Dic53—81no—
160#3230—531no—
161#3209—531no—
162#2868—531no—
163#1994—531no—
164#1549—531no—
165#1548—531no—
166#1276D5311yescharacteristic
167C2×C53—22no—
168C2×C53#200211yescharacteristic
169C2×C53#200511yescharacteristic
170C2×C53#201611yescharacteristic
171#533—531no—
172C2×C4⋊Q8—531no—
173C2×C4⋊Q8—531no—
174C2×C4⋊Q8—1062no—
175#500—1061no—
176#494—1061no—
177#487—1061no—
178#487—1061no—
179#484—1062no—
180#477—1062no—
181#462—1062no—
182#451—1062no—
183#431C2×D5311yescharacteristic
184#431C2×D5311yescharacteristic
185#423—531no—
186#384—1062no—
187#384—1062no—
188C43C2×D5311yescharacteristic
189#339—1062no—
190#333—1062no—
191#332—1062no—
192C53#1029311yescharacteristic, complemented, Sylow
193C22×Q8—532no—
194C22×Q8—1061no—
195C2×Q16—1064no—
196C2×M16—1061no—
197C4⋊Q8—531no—
198C4⋊Q8—531no—
199C4⋊Q8—1061no—
200C4⋊Q8—1061no—
201C4⋊Q8—1061no—
202C4⋊Q8—1061no—
203C4⋊Q8—1061no—
204C4⋊Q8—1061no—
205C4⋊Q8—1061no—
206C4⋊Q8—1061no—
207C4⋊Q8—2128no—
208#125—1061no—
209C22⋊Q8—1061no—
210C22⋊Q8—1061no—
211C4×Q8—1061no—
212C4×Q8—1061no—
213#117—1061no—
214C2×C4⋊C4—532no—
215C2×C4⋊C4—1068no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42D21212yes—
222C2×C42C22×D5311yescharacteristic
223#107—1064no—
224C4⋊C8—1061no—
225C4⋊C8—1061no—
226C4⋊C8D21212yes—
227C4⋊C8D21212yes—
228#103—1064no—
229#103—1064no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C2×Q8—1061no—
235C2×Q8—1062no—
236C2×Q8—1061no—
237C2×Q8—1062no—
238C2×Q8—2124no—
239C2×Q8—2124no—
240C2×Q8—2121no—
241C2×Q8—2121no—
242C22×C4—22no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4C2×D21211yescharacteristic
246C22×C4D4×D5311yescharacteristic
247C22×C4D4×D5311yescharacteristic
248Q16—4244no—
249M16—2122no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—2121no—
253C4⋊C4—1061no—
254C4⋊C4—1062no—
255C4⋊C4—1061no—
256C4⋊C4—1062no—
257C4⋊C4—1061no—
258C4⋊C4—1061no—
259C4⋊C4—21216no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C22⋊C4—2121no—
269C42—24no—
270C42—24no—
271C42—22no—
272C42—2121no—
273C42Dic21214yescomplemented
274C42C2×D21211yescharacteristic
275C42C2×D21211yescharacteristic
276C42D4×D5311yescharacteristic
277C42#1171994911yescharacteristic
278C23#1313318711yescharacteristic
279Q8—4242no—
280Q8—4242no—
281Q8—4242no—
282Q8—2122no—
283Q8—2124no—
284Q8—2122no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—2124no—
291C2×C4—2124no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2122no—
295C2×C4—2122no—
296C2×C4#1313318711yescharacteristic
297C2×C4#1313318711yescharacteristic
298C2×C4#1313319211yescharacteristic
299C2×C4#1313319211yescharacteristic
300C2×C4#1313319211yescharacteristic
301C2×C4#1313321411yescharacteristic
302C2×C4C2×Dic21212yes—
303C2×C4#1313325012yes—
304C8—42no—
305C8—42no—
306C8—4241no—
307C22—21no—
308C22—21no—
309C22#1681053311yescharacteristic
310C22#1681078112yes—
311C4—28no—
312C4—21no—
313C4—21no—
314C4—4242no—
315C4—4242no—
316C4—4241no—
317C4—4241no—
318C4#1681079212yes—
319C4#1681080712yes—
320C4#1681083412yes—
321C4#1681100311yescharacteristic
322C4#1681101711yescharacteristic
323C2—22no—
324C2#2027573311yescharacteristic
325C2#2027575411yescharacteristic
326C2#2027581711yescharacteristic
subgroupG/H
centerC22#16810533
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#1276D53
hypocenterC53#10293
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
89. C42×C53Q16semidirect
192. C53#10293semidirect
273. C42Dic212semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202858682
#202856652
#202831132
#202756922
#202733992
#202733952
C53×#12762
#1029353